LAUSR

In this paper, we introduce a space of $$\theta $$ -admissible distributions denoted by $${\mathcal {A}}_\theta ^*$$ as well as the notion of $$\theta $$ -admissible operators. We study the regularity properties of the classical conditional expectation acting on $${\mathcal {A}}_\theta ^*$$ and acting on $${\mathcal {L}}({\mathcal {A}}_\theta ,{\mathcal {A}}_\theta ^*)$$ which is the space of linear continuous operators from $${\mathcal {A}}_\theta $$ into $${\mathcal {A}}_\theta ^*$$ . An integral representation with respect to the coordinate system of the quantum white noise (QWN) derivatives and their adjoints $$\{D_t^{\pm }, D_t^{\pm *},\,t\in {\mathbb {R}}\}$$ of such conditional expectation is given. Then, we give a quantum white noise counterpart of the Clark formula. Finally, we introduce the QWN Hitsuda–Skorokhod integrals. Such integrals are shown to be QWN martingales using a new notion of QWN conditional expectation.